nLab
Lat

$\mathrm{Lat}$ is the category whose objects are lattices and whose morphisms are lattice homomorphisms, that is functions which preserve finitary meets and joins (equivalently, binary meets and joins and the top and bottom elements). $\mathrm{Lat}$ is a subcategory of Pos.

$\mathrm{Lat}$ is given by a finitary variety of algebras, or equivalently by a Lawvere theory, so has all the usual properties of such categories. By general abstract nonsense, the free lattice on a set $X$ exists, but it is not easy to describe in general.

• The free lattice on the empty set is the Boolean domain $\{\perp ,\top \}$.
• The free lattice on a singleton $\{x\}$ is the $3$-element totally ordered set $\{\perp ,x,\top \}$.
• The free lattice on $\{x,y\}$ is (if I haven't messed up) $\{\perp ,x\wedge y,x,y,x\vee y,\top \}$.
• The free lattice on $3$ generators, despite the simplicity of the previous examples, is infinite.
• The free lattice on any countable set can be made into a sublattice of the free lattice on $3$ generators!
• The free lattice on an uncountable set is uncountable, of course.

The results about the free lattice on $3$ generators may be found in Section 4.6 of Stone Spaces, which sites Skornjakov's Elements of Lattice Theory.

category: category